Constructive characterizations of 3-connected matroids of path width three

Abstract

A matroid M is sequential or has path width 3 if M is 3-connected and its ground set has a sequential ordering, that is, an ordering ( e 1 , e 2 , … , e n ) such that ( { e 1 , e 2 , … , e k } , { e k + 1 , e k + 2 , … , e n } ) is a 3-separation for all k in { 3 , 4 , … , n − 3 } . This paper proves that every sequential matroid is easily constructible from a uniform matroid of rank or corank two by a sequence of moves each of which consists of a slight modification of segment-cosegment or cosegment-segment exchange. It is also proved that if N is an n -element sequential matroid, then N is representable over all fields with at least n − 1 elements; and there is an attractive family of self-dual sequential 3-connected matroids such that N is a minor of some member of this family.